課程名稱︰電磁學
課程性質︰物理系必修
課程教師︰賀培銘
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰2011/11/10
考試時限（分鐘）：180 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
若答案完全正確，沒有計算過程不扣分；但答案未化約到最簡形式會酌量扣分
1.
(a) (5%) What is the volume charge density ρ(r) of an infinitely long
cylinder of radius R with a constant surface charge density σ ? The central
axis of the cylinder with the z-axis. Express your answer in cylindrical
coordibates.
(b) (10%) Find A,B in the following identity
2 2
d d d
x ----- δ(x-a) = A ---- δ(x-a) + B --- δ(x-a)
2 2 dx
dx dx
2. (15%) A solid ball of radius a with total charge q is covered by a
concentric spherical conducting shell from radius r = a to b. The total
charge on the conductor is Q, and the volume charge density is constant
from r = 0 to a. (圖就是兩個同心圓，小的半徑a，大的半徑b)
(a) Fibd the total charge at the surface at r= b.
(b) Find the electric potential V(r) in all 3 regions
(i) r > b , (ii) b > r > a , (iii) a > r.
3. (15%) An infinitely long rectangular metal pipe (sides a & b) extends
along the z direction. The metal plates at x = ±a/2 are grounded,
V(±a/2,y) = 0, and that at y = ±b/2 is given by the function
V(x, ±b/2) = A sin(2πx/a) for a given constant A.
(a) Find V(x,y) inside the pipe.
(Hint:{sin(nπ(x + a/2)/a} n=1~∞
is a complete basis of functions on the domain [-a/2 , a/2]. )
(b) Find the charge density σ(x) at the boundary y = (b/2)^-
(on the inner surface of the conductor).
圖： XY截面上的長方形，四邊：x=±a/2 , y=±b/2
4. (15%) The multipole expansion of the electric potential is
^ ^ ^
1 Q p。r r_i r_j Q_ij
V(r) = ------- [ ---- + ------ + Σ ------------ + ....]
4πε_0 r r^2 i,j 2 r^3
where Q_ij = ∫dτ' ρ(r') [3 r_i' r_j' - (r')^2 δ_ij]
For the charge distribution of 2 point charges of equal charge q at
(x,y,z) = (1,0,0),(-1,0,0), find
(1) its monopole, dipole and quadrupole moments,
(2) an approximate value of the potential V(0,0,z) on the z axis for
large z , ignoring terms that are of order 1/z^4 or smaller.
5. (10%) A point charge q is placed at the center (r=0) of a spherical
region of radius R. The potential V(R,θ) is given on the boundary and
1
we treat the integrals ν_ι =∫ V(R,θ)P_ι(cosθ)dcosθ as given numbers.
-1
Find the electric potential V(r,θ) for r > R.
6. (30%) Which of the following statements is(are) correct?
(a) For an arbitrary function f(x) defined on the region [0,π], Fourier's
trick allows us to ffind coeddicients f_n such that f(x) equals the sum
∞
Σ f_n sin(nx) for all x ∈(0,π) even when f(x) has discontinuity.
n=1
(b) The total charge on a conductor is always 0 because the charge density
ρ = 0 throughout the conductor.
(c) If we hide a charge q inside a conducting shell, there is no way to find
the value of q by measuring the electric field outside the conducting shell.
(d) To solve the electric potential for a point charge q inside a grounded
metal box, we only need 6 image charge , one for each side of the box.
(e) A×B is a vector field if A is a vector field and B is a pseudo vector
field.
(f) If ▽。F = 0 for a vector field F, there always exsits a scalar field
U such that F = - ▽U.
(g) The electric energy for a system of point charges is always positive
because W = ε/2 ∫E^2 dτ and E^2 is always positive.
(h) Across a boundary with surface charge density σ≠0, there is always a
discontinuity in the electric potential V.
(i) The followinf integral vanishes.
^
r
∫ ▽。(----) dτ= 0
All space r^2
(j) The capacitance of a capacitor is defined by C=Q/V and its electric
energy is given by W = 1/2 V^2 / C .
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